When a line divides an additional line segment right into two same halves through its midpoint in ~ 90º, that is called the perpendicular of the line segment. The perpendicular bisector theorem states that any suggest on the perpendicular bisector is equidistant native both the endpoints of the line segment on which the is drawn. If a pillar is standing at the center of a leg at one angle, all the points on the shaft will it is in equidistant from the finish points that the bridge.
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|1.||What is a Perpendicular Bisector?|
|2.||What is Perpendicular Bisector Theorem?|
|3.||What is the Converse the Perpendicular Bisector Theorem?|
|4.||Proof that Perpendicular Bisector Theorem|
|5.||Solved instances on Perpendicular Bisector Theorem|
|6.||Practice questions on Perpendicular Bisector Theoerem|
|7.||FAQs top top Perpendicular Bisector Theorem|
What is a Perpendicular Bisector?
A perpendicular bisector is a heat segment that intersects another line segment at a ideal angle and it divides that various other line into two equal components at that is midpoint.
What is Perpendicular Bisector Theorem?
The perpendicular bisector theorem says that any suggest on the perpendicular bisector is equidistant indigenous both the endpoints that the line segment ~ above which that is drawn.
In the above figure,
MT = NT
MS = NS
MR = NR
MQ = NQ
What is the Converse the Perpendicular Bisector Theorem?
The converse that the perpendicular bisector theorem states that if a allude is equidistant from both the endpoints of the line segment in the exact same plane, then that suggest is top top the perpendicular bisector the the heat segment.
In the over image, XZ=YZ
It indicates ZO is the perpendicular bisector that the heat segment XY.
Proof that Perpendicular Bisector Theorem
Let us look at the proof of the above two theorems on a perpendicular bisector.
Perpendicular Bisector organize Proof
Consider the adhering to figure, in which C is an arbitrary point on the perpendicular bisector of AB (which intersects AB at D):
Compare (Delta ACD) and (Delta BCD). We have:AD = BDCD = CD (common)∠ADC =∠BDC = 90°
We see that (Delta ACD cong Delta BCD) by the SAS congruence criterion. CA = CB,which means that C is equidistant native A and also B.
Note: refer to the SAS congruence criterion to know why (Delta ACD) and (Delta BCD) are congruent.
Perpendicular Bisector organize Converse Proof
Consider CA = CB in the over figure.
To prove that ad = BD.
Draw a perpendicular heat from suggest C that intersects line segment abdominal muscle at suggest D.
Now, compare (Delta ACD) and (Delta BCD). Us have:AC= BCCD = CD(common)∠ADC = ∠BDC = 90°
We check out that (Delta ACD cong Delta BCD) by the SAS congruence criterion. Thus, advertisement = BD, which method that C is equidistant indigenous A and B.
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Important NotesThe perpendicular bisector theorem and its converse have the right to be verified by the SAS congruency criterion.The perpendicular bisector to organize is offered in the building and construction of buildings, bridges, etc., and in do designs whereby we require to construct something in the center and also at equal street from the endpoints.
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